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0.5setgray0
0.5setgray1 FRIDAY:
The Banks Casher RelationKim Splittorff
What The QCD Phase Diagram & Dirac Spectra
Activity The Banks Casher relation (µ = 0)
What The Banks Casher relation (µ 6= 0)
The Sign Problem
Why Understand unquenched lattice QCD
FRIDAY:The Banks Casher Relation – p.1/37
QCD Phase diagram & Dirac Spectra
T
T
Chiral symmetry broken
Pions Protons Neutrons ...
Chirally symmetricQuarks Gluons
c
QCD
µ
1st oder
FRIDAY:The Banks Casher Relation – p.2/37
The sign problema problem for lattice QCD
ZNf=2 =
∫
dA det2(iD + µγ0 +m) e−SYM
Anti Hermitian Hermitian
det2(iDηγη + µγ0 +m) = |det(iDηγη + µγ0 +m)|2e2iθ
The measure is not real and positive
FRIDAY:The Banks Casher Relation – p.3/37
Yesterday: Why don’t we just quench the Determinant ?T
µm /2 m /3π N
Bose Einstein phase transition inQuenched QCD
Today: What is the effect of the Determinant ?How difficult is it to include it ?
OFRIDAY:The Banks Casher Relation – p.4/37
Yesterday: Why don’t we just quench the Determinant ?T
µm /2 m /3π N
Bose Einstein phase transition inQuenched QCD
Today: What is the effect of the Determinant ?How difficult is it to include it ?
FRIDAY:The Banks Casher Relation – p.4/37
Activity: Banks Casher relation at µ = 0
FRIDAY:The Banks Casher Relation – p.5/37
What is the effect of the Determinant ?
FRIDAY:The Banks Casher Relation – p.6/37
Quark mass & the eigenvalue distribution
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Re(λ)
Im(λ)
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
-0.3 -0.2 -0.1 0 0.1 0.2 0.3
Re(λ)
Im(λ)
X
quark mass
inside
X
quark mass
outside
µ < mπ/2 µ > mπ/2
(iDηγη + µγ0)ψk = zkψk det(iDηγη + µγ0 +m) =∏
k
λk +m
Bloch Wettig Lattice 2006 Gibbs PRINT-86-0389
Davies Klepfish PLB 256 (1991) 68 Lombardo Kogut Sinclair PRD 54 (1996) 2303
FRIDAY:The Banks Casher Relation – p.7/37
Electrostatic analogy suggests
Im(z)
Re(z)X
X
X
X
X
X
X
X
X
X
m
< >=0
< >(m)
ψ ψ
ψ ψ_
_
FRIDAY:The Banks Casher Relation – p.8/37
The big pictureT
µm /2 m /3π N
inside eigenvaluesQuark mass
NOT a phase transition in ZNf
Chiral condensate is independent of µ at T = 0 and µ < mN/3
OFRIDAY:The Banks Casher Relation – p.9/37
The big pictureT
µm /2 m /3π N
inside eigenvaluesQuark mass
NOT a phase transition in ZNf
Chiral condensate is independent of µ at T = 0 and µ < mN/3
FRIDAY:The Banks Casher Relation – p.9/37
The Silver Blaze Problem
Sir Arthur Conan Doyle The Memoirs of Sherlock Holmes: Silver Blaze
Thomas D . Cohen PRL (2003) 222001
FRIDAY:The Banks Casher Relation – p.10/37
µ = 0 Banks Casher
X
X
X
X
X
X
X
X
Re(z)
Im(z)
< >
< >(m)
ψ ψ
ψ ψ_
_
m
〈ψψ〉 =π
Vρ(0)
Banks Casher NPB 169 (1980) 103
FRIDAY:The Banks Casher Relation – p.11/37
µ 6= 0 The silver blaze problemIm(z)
Re(z)X
X
X
X
X
X
X
X
X
X
< >
< >(m)
ψ ψ
ψ ψ_
_
m
Eigenvalues move into the complex planethe discontinuity of the chiral condensate remains
Barbour et al. NPB 275 (1986) 296Gibbs PLB 182 (1986) 369
Cohen PRL 91 (2003) 222001FRIDAY:The Banks Casher Relation – p.12/37
We need Microscopic-regime of QCDThe eigenvalue density z〈ψψ〉 1√
V
SBχS The basic assumption
Chiral limit m〈ψψ〉 1√V
Small chemical potential µ2F 2π 1√
V
Notice µ ∼ mπ
Gasser, Leutwyler, PLB 184 (1987) 83, PLB 188 (1987) 477Neuberger, PRL 60 (1988) 889
Leutwyler, Smilga, PRD 46 (1992) 5607Shuryak, Verbaarschot, NPA 560 (1993) 306
Stephanov PRL 76 (1996) 4472Akemann PRL 89 (2002) 072002, J.Phys. A36 (2003) 3363
Splittorff, Verbaarschot, NPB 683 (2004) 467Osborn PRL 93 (2004) 222001
Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287
FRIDAY:The Banks Casher Relation – p.13/37
The unquenched eigenvalue densitym〈ψψ〉V = 100 increasing 2µ2F 2
πV
-1000100
-100
-50
0
50
100
-0.001
0
0.001
0.002
0.003
0.004
-1000100
-100
-50
0
50
100
-1000100
-100
-50
0
50
100
-0.001
0
0.001
0.002
0.003
0.004
-1000100
-100
-50
0
50
100
y〈ψψ〉V
Re[ρNf =1(x,y,m;µ)]
〈ψψ〉2V 2
2µ < mπ 2µ > mπ
x〈ψψ〉Vx〈ψψ〉V
For 2µ > mπ the density is complex and oscillatesOsborn PRL 93 (2004) 222001
Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287FRIDAY:The Banks Casher Relation – p.14/37
Definition of the eigenvalue density
Eigenvalue equation
(iDηγη + µγ0)ψj = zjψj
Eigenvalue density
ρNf (z, z∗,m;µ) ≡
*
X
j
δ2(z − zj)
+
QCD
〈O〉QCD ≡
R
dA O det(iDηγη + µγ0 +m)Nf e−SYM(A)
R
dA det(iDηγη + µγ0 +mf )Nf e−SYM(A)
OFRIDAY:The Banks Casher Relation – p.15/37
Definition of the eigenvalue density
Eigenvalue equation
(iDηγη + µγ0)ψj = zjψj
Eigenvalue density
ρNf (z, z∗,m;µ) ≡
*
X
j
δ2(z − zj)
+
QCD
〈O〉QCD ≡
R
dA O det(iDηγη + µγ0 +m)Nf e−SYM(A)
R
dA det(iDηγη + µγ0 +mf )Nf e−SYM(A)
FRIDAY:The Banks Casher Relation – p.15/37
The unquenched eigenvalue densitym〈ψψ〉V = 60 and 2µ2F 2
πV = 124
50 100 150-100
-50
0
50
100
-0.002-0.001
00.001
0.002-100
-50
0
50
100
Oscillations: Period ∼ 1/V
Amplitude ∼ e+V
Osborn PRL 93 (2004) 222001
Akemann Osborn Splittorff Verbaarschot NPB 712 (2005) 287
FRIDAY:The Banks Casher Relation – p.16/37
The chiral condensate from the eigenvalue density
〈ψψ〉(m) =1
V∂m logZ(m;µ)
=1
V
∫
dxdy ρ(x, y)1
x+ iy +m
The oscillations of the density areresponsible for chiral symmetry breaking
Osborn Splittorff Verbaarschot PRL 94 (2005) 202001
Ravagli Verbaarschot arXiv:0704.1111
FRIDAY:The Banks Casher Relation – p.17/37
The unquenched eigenvalue density
Structure: ρNf=1 = ρQ + ρU
Osborn, Splittorff, Verbaarschot hep-lat/0510118
FRIDAY:The Banks Casher Relation – p.18/37
The unquenched chiral condensate
−160 −120 −80 −40 0 40 80 120 160
−1.2
−0.9
−0.6
−0.3
0
0.3
0.6
0.9
1.2
Σ(m
ΣV)
−160 −120 −80 −40 0 40 80 120 160
−1.2
−0.9
−0.6
−0.3
0
0.3
0.6
0.9
1.2
Σ Q(m
ΣV,µ
FV
1/2 )
−160 −120 −80 −40 0 40 80 120 160mΣV
−1.2
−0.9
−0.6
−0.3
0
0.3
0.6
0.9
1.2
Σ U(m
ΣV,µ
FV
1/2 )
Structure: ΣNf=1(m) = ΣQ(m) + ΣU (m)
Splittorff hep-lat/0610072
FRIDAY:The Banks Casher Relation – p.19/37
Banks-Casher µ = 0
Accumulation of eigenvalues on the y-axis isresponsible for chiral symmetry breaking
New mechanism µ 6= 0
The oscillations of the eigenvalue density areresponsible for chiral symmetry breaking
FRIDAY:The Banks Casher Relation – p.20/37
How to calculate the eigenvalue density
FRIDAY:The Banks Casher Relation – p.21/37
The replica way of writing the eigenvalue density
ρNf (z, z∗,m;µ) = limn→0
1
πn∂z∗∂z logZNf ,n(m, z, z∗;µ)
generating functionals for the eigenvalue density
ZNf ,n(m, z, z∗;µ) =∫
dA det(iDηγη + µγ0 +m)Nf | det(iDηγη + µγ0 + z)|2n e−SYM(A)
Stephanov PRL 76 (1996) 4472
FRIDAY:The Banks Casher Relation – p.22/37
Central observation
The eigenvalue z and its complex conjugate z∗ appears as the mass of
two conjugate fermions.
very small eigenvalues ↔ very light quarks
⇒ Compton wavelength of the pions boxsize
⇒ Zero mode of the pions dominates
ZNf ,n =
∫
U(Nf+2n)
dUe−V4F 2πµ
2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)
OFRIDAY:The Banks Casher Relation – p.23/37
Central observation
The eigenvalue z and its complex conjugate z∗ appears as the mass of
two conjugate fermions.
very small eigenvalues ↔ very light quarks
⇒ Compton wavelength of the pions boxsize
⇒ Zero mode of the pions dominates
ZNf ,n =
∫
U(Nf+2n)
dUe−V4F 2πµ
2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)
OFRIDAY:The Banks Casher Relation – p.23/37
Central observation
The eigenvalue z and its complex conjugate z∗ appears as the mass of
two conjugate fermions.
very small eigenvalues ↔ very light quarks
⇒ Compton wavelength of the pions boxsize
⇒ Zero mode of the pions dominates
ZNf ,n =
∫
U(Nf+2n)
dUe−V4F 2πµ
2Tr[U,B][U−1,B] + 12m〈ψψ〉V Tr(U+U−1)
FRIDAY:The Banks Casher Relation – p.23/37
The replica way of writing the eigenvalue density
ρNf (z, z∗,m;µ) = limn→0
1
πn∂z∗∂z logZNf ,n(m, z, z∗;µ)
generating functionals for the eigenvalue density
ZNf ,n(m, z, z∗;µ) =∫
dA det(iDηγη + µγ0 +m)Nf | det(iDηγη + µγ0 + z)|2n e−SYM(A)
Stephanov PRL 76 (1996) 4472
FRIDAY:The Banks Casher Relation – p.24/37
The Replica Limit of the Toda Lattice Equation
∂z∂z∗ logZNf ,n = 4zz∗nZNf ,n+1ZNf ,n−1
[ZNf ,n]2
Take n→ 0 in this equation
ρNf(z, z∗,m;µ) = 4zz∗
ZNf ,n=1(m, z, z∗;µ)ZNf ,n=−1(m|z, z∗;µ)
[ZNf(m;µ)]2
ProblemsVerbaarschot, Zirnbauer, J. Phys. A 18, 1093 (1985)
Kamenev Mézard J.Phys.A 32 4373 (1999); PRB 60 3944 (1999)Yurkevich, Lerner, PRB 60, 3955 (1999)
M.R. Zirnbauer, cond-mat/9903338Solution
Kanzieper, PRL 89, 250201 (2002)
Splittorff, Verbaarschot, PRL 90, 041601 (2003)FRIDAY:The Banks Casher Relation – p.25/37
Today: What is the effect of the Determinant ? (DONE)How difficult is it to include it ? (NOW)
FRIDAY:The Banks Casher Relation – p.26/37
The Big Picture
T
µm /3N
B Sχ
m /2π
Mon
te C
arlo
No Monte Carlo
Lattice Pioneers
FRIDAY:The Banks Casher Relation – p.27/37
The lattice Pioneers
4.84.824.844.864.884.9
4.924.944.964.98
55.025.045.06
0 0.5 1 1.5 2
1.0
0.95
0.90
0.85
0.80
0.75
0.70
0 0.1 0.2 0.3 0.4 0.5
β
T/T
c
µ/T
a µ
confined
QGP<sign> ~ 0.85(1)
<sign> ~ 0.45(5)
<sign> ~ 0.1(1)
D’Elia, Lombardo 163
Azcoiti et al., 83
Fodor, Katz, 63
Our reweighting, 63
This work, 63
de Forcrand Philipsen JHEP PoS LAT2005 (2005) 016
FRIDAY:The Banks Casher Relation – p.28/37
The average phase factor
〈sign〉 = 〈e2iθ〉 ≡
⟨
det(D + µγ0 +m)
det(D + µγ0 +m)∗
⟩
is a ratio of two partition functions
〈e2iθ〉1+1∗ =Z1+1
Z1+1∗
FRIDAY:The Banks Casher Relation – p.29/37
The average phase factor in the microscopic limit
〈e2iθ〉1+1∗ =Z1+1
Z1+1∗
=I0(m)2 − I1(m)2
2e2µ2∫ 1
0dtte−2µ2t2I0(mt)2
0 0.5 1 1.52µ/mπ
0
0.2
0.4
0.6
0.8
1
<ex
p(2i
θ)>
1+1*
mΣV = 1 mΣV >> 1
Splittorff Verbaarschot PRL 98 (2007) 031601FRIDAY:The Banks Casher Relation – p.30/37
The average phase factor
〈sign〉 = 〈e2iθ〉 ≡
⟨
det(D + µγ0 +m)
det(D + µγ0 +m)∗
⟩
is a ratio of two partition functions
〈e2iθ〉1+1∗ =Z1+1
Z1+1∗= e−V∆Ω
Phase transition at µ = mπ/2
FRIDAY:The Banks Casher Relation – p.31/37
The big pictureT
µm /2 m /3π N
inside eigenvaluesQuark mass
Phase transition in Z1+1∗
FRIDAY:The Banks Casher Relation – p.32/37
Lattice measurements of the endpoint
0 0.5 1 1.5 22µ/mπ
0
0.2
0.4
0.6
0.8
1
1.2
T/T
0
Fodor Katz JHEP 0203 (2002) 014, JHEP 0404 (2004) 050
Splittorff PoS LAT2006:023
FRIDAY:The Banks Casher Relation – p.33/37
The current questionT
µm /2 m /3π N
inside eigenvaluesQuark mass
Sign problem ?
FRIDAY:The Banks Casher Relation – p.34/37
Summary
FRIDAY:The Banks Casher Relation – p.35/37
Equivalence chRMT and QCD through low energy effective theory
Mean field chRMT & QCD Phase Diagram
Microscopic spectral correlation functions
The sign problem
OFRIDAY:The Banks Casher Relation – p.36/37
Equivalence chRMT and QCD through low energy effective theory
Mean field chRMT & QCD Phase Diagram
Microscopic spectral correlation functions
The sign problem
OFRIDAY:The Banks Casher Relation – p.36/37
Equivalence chRMT and QCD through low energy effective theory
Mean field chRMT & QCD Phase Diagram
Microscopic spectral correlation functions
The sign problem
OFRIDAY:The Banks Casher Relation – p.36/37
Equivalence chRMT and QCD through low energy effective theory
Mean field chRMT & QCD Phase Diagram
Microscopic spectral correlation functions
The sign problem
FRIDAY:The Banks Casher Relation – p.36/37
Thanks !!
FRIDAY:The Banks Casher Relation – p.37/37